Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #356055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22⋊C42 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2443 conjugacy classes of subgroups; only the 472 normal subgroups are shown.
This group has 2443 conjugacy classes of subgroups — too many to draw. Only its 472 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15,17,8)(5,9)(6,10)(7,14)(12,20,27,25)(13,23)(16,28,30,22)(18,21)(19,24)(26,31,29,32);(1,3)(2,7,10,11)(4,12)(5,18,19,13)(6,14)(8,20,15,25)(9,23)(16,29,30,26)(17,27)(21,24)(22,32)(28,31);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,25)(13,26)(14,27)(18,29)(19,30)(21,31)(23,32)(24,28);(1,5)(2,9)(3,13)(4,16)(6,19)(7,21)(8,22)(10,24)(11,23)(12,26)(14,18)(15,28)(17,30)(20,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=be,c^a=cf,f^a=fi,h^a=hi,d^b=de,e^b=ei,g^c=gi,h^d=hi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72210444, #72210507, #72210526, … |
| maximal normal subgroup of | #72210444, #72210507, #72210526, … |
| maximal quotient of | #992, C2×#356055 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263251);